Geometric Programming and its Application in Network Resource Allocation. Presented by: Bin Wang

Transcription

1 Geometric Programming and its Application in Network Resource Allocation Presented by: Bin Wang

2 Why this talk? Nonlinear and nonconvex problem, can be turned into nonlinear convex problem Global optimal, zero dual gap, Numerical efficiency: interior point (polynomial) Wide range of application Circuit design Signal processing and information theory Network resource allocation Not just using software package and applying some algorithms, it involves skills in modeling and approximation

3 Outlines Basic Geometric Programming Extended Geometric Programming References

4 Outlines Basic Geometric Programming What is GP? Example (Power Control, rate allocation) Extended Geometric Programming References

5 What is GP?

6 What is GP? Limitation: Objective must be the minimization of posynomial Equality constraints can only have the form of a monomial equal one inequality constraints can only have the form of a posynomial less than or equal to one Nonlinear and nonconvex

7 What is GP? The trick to solve GP problem is to use original variables logarithms:, this results in the standard problem now becomes: GP convex form

8 Example

9 GP example- power control CDMA based ad-hoc network, it has n transmitters, labeled 1,..., n, which transmit at (positive) power levels P1,..., Pn, which are the variables in our power control problem. We also have n receivers, labeled 1,..., n; The power received from transmitter j, at receiver i, is given by GijPj. The signal power at receiver i is GiiPi, and the interference power at receiver i is. The noise power at receiver i is given by. The signal to interference and noise ratio (SINR) of the ith receiver/transmitter pair is given by, we require that We also impose limits on the transmitter powers:

10 GP application 1: power control The problem of minimizing the total transmitter power, subject to these constraints, can be expressed as This is not a GP, but is easily cast as a GP, by taking the inverse of the SINR constraints:

11 GP application 2: Rate allocation Consider the same CDMA based ad-hoc network, system objective is to achieve the maximization of the proportional fairness rate allocation : Since And maximize is equal to minimize

12 Outlines Basic Geometric Programming Extended Geometric Programming Extended Geometric Programming GP Approximation References

13 Extensions Simple transformation the inverse of the SINR constraints Maximized objectives to minimization problem Generalized GP that allows compositions of posynomials with other functions Use techniques: introduce new variable GP formulations based on monomial approximations of nonlinear functions. Objective or constraints can not be transformed into GP problems

14 Example: Monomial Approximation If we can approximate the denominator with a monomial, but leaving the numerator unchanged Let, then we use here, for any fixed positive, is the best local monomial approximation to near

15 Monomial approximation Asking for a monomial approximation of f (x) near x corresponds to ask for an affine approximation of Such an approximation is provided by the first order Taylor approximation of F: (*) Using this formula, and taking the exponential of (*), we have:

16 Iterative Approximation Iterative approximation:

17 Monomial approximation Any heuristics? How about the performance? With the guarantee of KKT condition, monomial approximation can get over 90% global optimal, with the left results to be local optimal.

18 References S.Boyd, L.Vandenberghe, Convex Optimization, Cambridge University Press (2004) M. Chiang, Geometric Programming for Communication Systems," Foundations and Trends in Communications and Information Theory, vol. 2, no. 1, pp , Aug M. Chiang, C. W. Tan, D.Palomar, D. O'Neill, and D. Julian Geometric programming for power control" IEEE Trans. Wireless Communications, R.Duffin, E.Peterson, C.Zener, Geometric Programming: Theory and Application, Wiley, New York (1967)